Probabilistic formulation of a performance-based matrix combining maximum and residual deformations

Reports from past earthquake reconnaissance observations as well as results from analytical studies indicate that most structures designed according to current codes will sustain residual (permanent) deformations in the event of a design-level earthquake, even if they perform exactly as expected. Despite this reality, little consideration is currently given to residual displacements unlike the maximum transient response during seismic design or performance assessment of structures. The concept of a performance-based matrix, where maximum and residual displacements are combined as complementary damage indicators has been recently introduced by the authors, is a viable tool to identify performance levels. In this paper, a probabilistic formulation of the performance based matrix concept is presented, within a refined framework for performance based seismic design and assessment of structures. Extensive non-linear time history analyses have been performed on equivalent SDOF systems designed using the Displacement Based Design (DBD) approach and modelled with different hysteresis rules to represent multi-storey reinforced concrete or steel buildings. Fragility curves representing the probabilities of exceedence of different joined maximum-residual performance levels are derived using bivariate probability distributions. Valuable observations are derived with regard to the contributions from the response parameters to the total probability of exceedence based on the hysteretic behaviour. As part of the proposed formulation for performance-based seismic design and assessment, performance objectives can be defined and associated with targeted probability of achieving them